The Stacks project

Lemma 52.28.4. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Assume

  1. $A$ is a graded ring, $\mathfrak a = A_+$, and $I$ is a homogeneous ideal,

  2. $(\mathcal{F}_ n) = (\widetilde{M_ n}|_ U)$ where $(M_ n)$ is an inverse system of graded $A$-modules, and

  3. $(\mathcal{F}_ n)$ extends canonically to $X$.

Then there is a finite graded $A$-module $N$ such that

  1. the inverse systems $(N/I^ nN)$ and $(M_ n)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-power torsion modules, and

  2. $(\mathcal{F}_ n)$ is the completion of of the coherent module associated to $N$.

Proof. Let $(\mathcal{G}_ n)$ be the canonical extension as in Lemma 52.16.8. The grading on $A$ and $M_ n$ determines an action

\[ a : \mathbf{G}_ m \times X \longrightarrow X \]

of the group scheme $\mathbf{G}_ m$ on $X$ such that $(\widetilde{M_ n})$ becomes an inverse system of $\mathbf{G}_ m$-equivariant quasi-coherent $\mathcal{O}_ X$-modules, see Groupoids, Example 39.12.3. Since $\mathfrak a$ and $I$ are homogeneous ideals the closed subschemes $Z$, $Y$ and the open subscheme $U$ are $\mathbf{G}_ m$-invariant closed and open subschemes. The restriction $(\mathcal{F}_ n)$ of $(\widetilde{M_ n})$ is an inverse system of $\mathbf{G}_ m$-equivariant coherent $\mathcal{O}_ U$-modules. In other words, $(\mathcal{F}_ n)$ is a $\mathbf{G}_ m$-equivariant coherent formal module, in the sense that there is an isomorphism

\[ \alpha : (a^*\mathcal{F}_ n) \longrightarrow (p^*\mathcal{F}_ n) \]

over $\mathbf{G}_ m \times U$ satisfying a suitable cocycle condition. Since $a$ and $p$ are flat morphisms of affine schemes, by Lemma 52.16.9 we conclude that there exists a unique isomorphism

\[ \beta : (a^*\mathcal{G}_ n) \longrightarrow (p^*\mathcal{G}_ n) \]

over $\mathbf{G}_ m \times X$ restricting to $\alpha $ on $\mathbf{G}_ m \times U$. The uniqueness guarantees that $\beta $ satisfies the corresponding cocycle condition. In this way each $\mathcal{G}_ n$ becomes a $\mathbf{G}_ m$-equivariant coherent $\mathcal{O}_ X$-module in a manner compatible with transition maps.

By Groupoids, Lemma 39.12.5 we see that $\mathcal{G}_ n$ with its $\mathbf{G}_ m$-equivariant structure corresponds to a graded $A$-module $N_ n$. The transition maps $N_{n + 1} \to N_ n$ are graded module maps. Note that $N_ n$ is a finite $A$-module and $N_ n = N_{n + 1}/I^ n N_{n + 1}$ because $(\mathcal{G}_ n)$ is an object of $\textit{Coh}(X, I\mathcal{O}_ X)$. Let $N$ be the finite graded $A$-module foud in Algebra, Lemma 10.98.3. Then $N_ n = N/I^ nN$, whence $(\mathcal{G}_ n)$ is the completion of the coherent module associated to $N$, and a fortiori we see that (b) is true.

To see (a) we have to unwind the situation described above a bit more. First, observe that the kernel and cokernel of $M_ n \to H^0(U, \mathcal{F}_ n)$ is $A_+$-power torsion (Local Cohomology, Lemma 51.8.2). Observe that $H^0(U, \mathcal{F}_ n)$ comes with a natural grading such that these maps and the transition maps of the system are graded $A$-module map; for example we can use that $(U \to X)_*\mathcal{F}_ n$ is a $\mathbf{G}_ m$-equivariant module on $X$ and use Groupoids, Lemma 39.12.5. Next, recall that $(N_ n)$ and $(H^0(U, \mathcal{F}_ n))$ are pro-isomorphic by Definition 52.16.7 and Lemma 52.16.8. We omit the verification that the maps defining this pro-isomorphism are graded module maps. Thus $(N_ n)$ and $(M_ n)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-power torsion modules. $\square$

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